×

Erdélyi-Kober fractional diffusion. (English) Zbl 1276.26021

Summary: The aim of this short note is to highlight that the generalized grey Brownian motion (ggBm) is an anomalous diffusion process driven by a fractional integral equation in the sense of Erdélyi-Kober, and for this reason here it is proposed to call such family of diffusive processes as Erdélyi-Kober fractional diffusion. The ggBm is a parametric class of stochastic processes that provides models for both fast and slow anomalous diffusion. This class is made up of self-similar processes with stationary increments and it depends on two real parameters: \(0 < \alpha \leq 2\) and \(0 < \beta \leq 1\). It includes the fractional Brownian motion when \(0 < \alpha \leq 2\) and \(\beta = 1\), the time-fractional diffusion stochastic processes when \(0 < \alpha = \beta < 1\), and the standard Brownian motion when \(\alpha = \beta = 1\). In the ggBm framework, the Mainardi function emerges as a natural generalization of the Gaussian distribution recovering the same key role of the Gaussian density for the standard and the fractional Brownian motion.

MSC:

26A33 Fractional derivatives and integrals
45D05 Volterra integral equations
60G22 Fractional processes, including fractional Brownian motion
33E30 Other functions coming from differential, difference and integral equations
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] A. Erdélyi, On fractional integration and its applications to the theory of Hankel transforms. Quart. J. Math. Oxford 11, No 1 (1940), 293-303. http://dx.doi.org/10.1093/qmath/os-11.1.293; · Zbl 0025.18602
[2] R. Gorenflo, Yu. Luchko, F. Mainardi, Analytical properties and applications of the Wright function. Fract. Calc. Appl. Anal. 2, No 4 (1999), 383-414; http://www.math.bas.bg/ fcaa; · Zbl 1027.33006
[3] R. Gorenflo, Yu. Luchko, F. Mainardi, Wright functions as scaleinvariant solutions of the diffusion-wave equation. J. Comput. Appl. Math. 118, No 1-2 (2000), 175-191. http://dx.doi.org/10.1016/S0377-0427(00)00288-0; · Zbl 0973.35012
[4] R. Gorenflo, F. Mainardi, Subordination pathways to fractional diffusion. Eur. Phys. J. Special Topics 193, (2011), 119-132. http://dx.doi.org/10.1140/epjst/e2011-01386-2;
[5] R. Gorenflo, F. Mainardi, Parametric subordination in fractional diffusion processes. In: Fractional Dynamics. Recent Advances, World Scientific, Singapore (2011), Chapter 10, 227-261.; · Zbl 1371.60135
[6] P. Grigolini, A. Rocco, B.J. West, Fractional calculus as a macroscopic manifestation of randomness. Phys. Rev. E 59, No 3 (1999), 2603-2613. http://dx.doi.org/10.1103/PhysRevE.59.2603;
[7] V. Kiryakova, Generalized Fractional Calculus and Applications. Longman Scientific & Technical and J. Wiley, Harlow - N. York (1994).;
[8] J. Klafter, I.M. Sokolov, Anomalous diffusion spreads its wings. Physics World, August (2005), 29-32.;
[9] H. Kober, On a fractional integral and derivative. Quart. J. Math. Oxford 11, No 1 (1940), 193-211. http://dx.doi.org/10.1093/qmath/os-11.1.193; · Zbl 0025.18502
[10] Yu. Luchko, Operational rules for a mixed operator of the Erdélyi-Kober type. Fract. Calc. Appl. Anal. 7, No 3 (2004), 339-364; http://www.math.bas.bg/ fcaa; · Zbl 1128.26004
[11] Yu. Luchko, J. Trujillo, Caputo-type modification of the Erdélyi-Kober fractional derivative. Fract. Calc. Appl. Anal. 10, No 3 (2007), 249-267; http://www.math.bas.bg/ fcaa; · Zbl 1152.26304
[12] B.N. Lundstrom, M.H. Higgs, W.J. Spain, A.L. Fairhall, Fractional differentiation by neocortical pyramidal neurons. Nature Neuroscience 11, No 11 (2008), 1335-1342. http://dx.doi.org/10.1038/nn.2212;
[13] F. Mainardi, Fractional relaxation-oscillation and fractional diffusion-wave phenomena. Chaos, Solitons & Fractals 7, No 9 (1996), 1461-1477. http://dx.doi.org/10.1016/0960-0779(95)00125-5; · Zbl 1080.26505
[14] F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity. Imperial College Press, London (2010). http://dx.doi.org/10.1142/9781848163300; · Zbl 1210.26004
[15] F. Mainardi, Y. Luchko, G. Pagnini, The fundamental solution of the space-time fractional diffusion equation. Fract. Calc. Appl. Anal. 4, No 2 (2001), 153-192; http://www.math.bas.bg/ fcaa; · Zbl 1054.35156
[16] F. Mainardi, A. Mura, G. Pagnini, The M-Wright function in timefractional diffusion processes: A tutorial survey. Int. J. Diff. Equations 2010, (2010), 104505.; · Zbl 1222.60060
[17] F. Mainardi, G. Pagnini, The role of the Fox-Wright functions in fractional sub-diffusion of distributed order. J. Comput. Appl. Math. 207, No 2 (2007), 245-257. http://dx.doi.org/10.1016/j.cam.2006.10.014; · Zbl 1120.35002
[18] F. Mainardi, G. Pagnini, R. Gorenflo, Mellin transform and subordination laws in fractional diffusion processes. Fract. Calc. Appl. Anal. 6, No 4 (2003), 441-459; http://www.math.bas.bg/ fcaa; · Zbl 1083.60032
[19] R. Metzler, J. Klafter, The restaurant at the end of the random walk: recent developments in fractional dynamics descriptions of anomalous dynamical processes. J. Phys. A: Math. Gen. 37, No 31 (2004), R161-R208. http://dx.doi.org/10.1088/0305-4470/37/31/R01; · Zbl 1075.82018
[20] A. Mura, Non-Markovian Stochastic Processes and Their Applications: From Anomalous Diffusion to Time Series Analysis. Ph.D. Thesis, University of Bologna (2008); http://amsdottorato.cib.unibo.it/846/1/TesiMuraAntonio.pdf, Now available by Lambert Academic Publishing (2011).;
[21] A. Mura, F. Mainardi, A class of self-similar stochastic processes with stationary increments to model anomalous diffusion in physics. Integr. Transf. Spec. Funct. 20, No 3 (2009), 185-198. http://dx.doi.org/10.1080/10652460802567517; · Zbl 1173.26005
[22] A. Mura, G. Pagnini, Characterizations and simulations of a class of stochastic processes to model anomalous diffusion. J. Phys. A: Math. Theor. 41, No 28 (2008), 285003. http://dx.doi.org/10.1088/1751-8113/41/28/285003; · Zbl 1143.82028
[23] A. Mura, M.S. Taqqu, F. Mainardi, Non-Markovian diffusion equations and processes: Analysis and simulations. Physica A 387, No 21 (2008), 5033-5064. http://dx.doi.org/10.1016/j.physa.2008.04.035;
[24] G. Pagnini, Nonlinear time-fractional differential equations in combustion science. Fract. Calc. Appl. Anal. 14, No 1 (2011), 80-93; http://www.springerlink.com/content/1311-0454/14/1/ http://dx.doi.org/10.2478/s13540-011-0006-8; · Zbl 1273.34013
[25] G. Pagnini, The evolution equation for the radius of a premixed flame ball in fractional diffusive media. Eur. Phys. J. Special Topics 193, (2011), 105-117. http://dx.doi.org/10.1140/epjst/e2011-01385-3;
[26] I. Podlubny, Fractional Differential Equations. Academic Press, San Diego (1999).; · Zbl 0924.34008
[27] A. Rocco, B.J. West, Fractional calculus and the evolution of fractal phenomena. Physica A 265, No 3-4 (1999), 535-546. http://dx.doi.org/10.1016/S0378-4371(98)00550-0;
[28] R.K. Saxena, G. Pagnini, Exact solutions of triple-order time-fractional differential equations for anomalous relaxation and diffusion I: The accelerating case. Physica A 390, No 4 (2011), 602-613. http://dx.doi.org/10.1016/j.physa.2010.10.012;
[29] E. Scalas, The application of continuous-time random walks in finance and economics. Physica A 362, No 2 (2006), 225-239. http://dx.doi.org/10.1016/j.physa.2005.11.024;
[30] W.R. Schneider, Grey noise. In: Stochastic Processes, Physics and Geometry, World Scientific, Teaneck (1990), 676-681.;
[31] W.R. Schneider, Grey noise. In: Ideas and Methods in Mathematical Analysis, Stochastics, and Applications, Vol. I, Cambridge University Press, Cambridge (1992), 261-282.;
[32] I.N. Sneddon, Mixed Boundary Value Problems in Potential Theory. North-Holland Publ., Amsterdam (1966).; · Zbl 0139.28801
[33] I.N. Sneddon, The use in mathematical analysis of the Erdélyi-Kober operators and some of their applications, In: Lect. Notes Math. 457, Springer-Verlag, New York (1975), 37-79.;
[34] I.N. Sneddon, The Use of Operators of Fractional Integration in Applied Mathematics. RWN — Polish Sci. Publ., Warszawa-Poznan (1979).; · Zbl 0556.44001
[35] I.M. Sokolov, A.V. Chechkin, J. Klafter, Distributed-order fractional kinetics. Acta Phys. Pol. B 35, No 4 (2004), 1323-1341.;
[36] J.A. Tenreiro Machado, And I say to myself: “What a fractional world!”. Fract. Calc. Appl. Anal. 14, No 4 (2011), 635-654; http://www.springerlink.com/content/1311-0454/14/4/ http://dx.doi.org/10.2478/s13540-011-0037-1; · Zbl 1273.37002
[37] B.M. Vinagre, I. Podlubny, A. Hernández, V. Feliu, Some approximations of fractional order operators used in control theory and applications. Fract. Calc. Appl. Anal. 3, No 3 (2000), 231-248; http://www.math.bas.bg/ fcaa; · Zbl 1111.93302
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.